Both the ferromagnetic and Néel phases have magnetic order. In the ferromagnetic phase, all the spins point in the same direction, which are characterized by the non-zero spin-spin correlations \(\langle S_i^zS_{i+n}^z\rangle \) , where \(\langle \cdots \rangle \)  represents the expectation value in the ground state; while in the Néel phase, all the spins at the nearest neighbor sites are aligned in the opposite directions, which are characterized by the non-zero Néel order parameter \((-1)^n\langle S_i^zS_{i+n}^z\rangle \)  [1]}. In the XY phases, the spins prefer to lie in the \(xy\) -plane rather than along \(z\) -direction [2]}. The XY phases are gapless, which leads to a power-law decaying spin-spin correlations \(\langle S_i^+S_{i+r}^-\rangle \)  and \(\langle (S_i^+)^2(S_{i+r}^-)^2\rangle \)  in the XY1 and XY2 phases respectively [3]}, [4]}, [5]}. Both the Haldane and large-D phases are gapful, which leads to an exponentially decaying spin–spin correlations [6]}, [7]}. But in the Haldane phase, although the Néel order parameter vanishes as the ground state is disordered, there is a hidden order in the Haldane phase characterized by de Nijs and Rommelse's nonlocal topological string order parameter \(-\langle S_i^z\exp (i\pi \sum _{k=i+1}^{j-1}S_k^z)S_j^z\rangle \)  [8]}, [9]}, when the system preserves the hidden \(Z_2\times Z_2\)  symmetry [2]}, [11]}, corresponding to the rotation by \(\pi \)  around the \(z\)  and \(x\)  axes respectively [1]}.
